On orbits of automorphism groups.
A subgroup of a finite group is said to be conjugate-permutable if for all . More generaly, if we limit the element to a subgroup of , then we say that the subgroup is -conjugate-permutable. By means of the -conjugate-permutable subgroups, we investigate the relationship between the nilpotence of and the -conjugate-permutability of the Sylow subgroups of and under the condition that , where and are subgroups of . Some results known in the literature are improved and...